[for simplicity "lam"="lambda" and "del"="delta"]

y''=lam*y, 0<x<l
y(0)+y'(0)=0, y(l)=0

I need to find all values of l>0 such that the largest eigenvalue lam=lam(max) is nonnegative. Also need to define lam(max) implicitly along with its corresponding eigenfunction

characteristic equation has roots r=+/- sqrt(lam)

and in the first case lam>0=del^2 --> r=del

this gives the general solution
y=C1*cosh(del*x)+C2*sinh(del*x)
y'=C1*del*sinh(del*x)+C2*del*cosh(del*x)

and thus
y(0)+y'(0)=0=C1+del*C2 --> del=-C1/C2
y(l)=0=C1*cosh(del*l)+C2*sinh(del*l) --> tanh(del*l)=-C1/C2= del

therefore del=tanh(del*l)

graphing y=del vs. y=tanh(del*l) yields a single positive eigenvalue, but I'm at a loss as to how to determine an implicit equation for lam(max)

I've tried graphing it as a function of two variables to find a local maximum but I'm not getting anything useful

this is Dr. Martin's class btw on the off-chance someone happens to see this

[Edited on January 31, 2011 at 4:55 PM. Reason : oops...wait a minute...]

[Edited on January 31, 2011 at 4:59 PM. Reason : nevermind. stupid algebra error throwing it all off. thanks for your time!]

1/31/2011 4:48:24 PM